Question

Bill and Hector are playing a game at an arcade. Bill starts the next round of the game with 10 points and earns 10 more points each round. Hector starts the next round with 100 points but loses 5 more points each round. Is there a time when they’ll have the same number of points?
The equation below represents the number of points, y, that Hector has after x rounds. Use the equation to complete the table with the number of points Hector has after 4, 6, and 8 rounds. You can solve on paper and then enter your answer on Zearn.
y=100−5x

Rounds (x) Points (y)
4
6
8
Excellent!
Next, graph the values you just found on the coordinate plane to show the relationship between the number of rounds Hector plays and the number of points he has.
Rounds (x) Points (y)
4 80
6 70
8 60
You got it!
Next, consider Bill. The equation below represents the number of points, y, that Bill has after x rounds. Use the equation to complete the table with the number of points Bill has after 2, 3, and 5 rounds. You can solve on paper and then enter your answer on Zearn.
y=10+10x

Rounds (x) Points (y)
2
3
5
Nice!
Here’s the line you already graphed for Hector, y=100−5x. Now, use the values that you just found for Bill to graph the relationship between the number of rounds he plays and the number of points he has on this same graph.
Rounds (x) Points (y)
2 30
3 40
5 60
Excellent!
The points you just plotted for Bill have been connected to form line B. What does the highlighted point, the point of intersection, represent in the context of this problem?
x
y
B
H
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
Rounds
Points
After
rounds, Bill and Hector both have
points.

Answer 1

Let's start with Hector's points using the equation \(y = 100 - 5x\).

1. **Calculating Hector's Points:**
- For \(x = 4\):
\[
y = 100 - 5(4) = 100 - 20 = 80
\]
- For \(x = 6\):
\[
y = 100 - 5(6) = 100 - 30 = 70
\]
- For \(x = 8\):
\[
y = 100 - 5(8) = 100 - 40 = 60
\]

2. **Hector's Points Table:**

| Rounds (x) | Points (y) |
|------------|------------|
| 4 | 80 |
| 6 | 70 |
| 8 | 60 |

Next, we will calculate Bill's points using the equation \(y = 10 + 10x\).

3. **Calculating Bill's Points:**
- For \(x = 2\):
\[
y = 10 + 10(2) = 10 + 20 = 30
\]
- For \(x = 3\):
\[
y = 10 + 10(3) = 10 + 30 = 40
\]
- For \(x = 5\):
\[
y = 10 + 10(5) = 10 + 50 = 60
\]

4. **Bill's Points Table:**

| Rounds (x) | Points (y) |
|------------|------------|
| 2 | 30 |
| 3 | 40 |
| 5 | 60 |

Now, let's summarize the points we would plot for both Bill and Hector.

5. **Hector's Points:**
- 4 Rounds: 80 Points
- 6 Rounds: 70 Points
- 8 Rounds: 60 Points

6. **Bill's Points:**
- 2 Rounds: 30 Points
- 3 Rounds: 40 Points
- 5 Rounds: 60 Points

Next, let's analyze the intersection point of their graphs.

To find the point at which they have the same number of points, we set the two equations equal to each other:

\[
10 + 10x = 100 - 5x
\]

Solving for \(x\):
\[
10 + 10x + 5x = 100 \\
15x = 100 - 10 \\
15x = 90 \\
x = 6
\]

Now substituting \(x = 6\) back into either equation to find \(y\):
\[
y = 10 + 10(6) = 10 + 60 = 70
\]
or
\[
y = 100 - 5(6) = 100 - 30 = 70
\]

Thus, the intersection point is \((6, 70)\).

7. **Final Statements:**
The highlighted point represents that after **6 rounds**, Bill and Hector will both have **70 points**.